Toward Effective Liouvillian Integration

We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension.